Physics · Ch 2 — Motion in a Straight Line

Acceleration

2.5

Acceleration

Just as velocity measures the rate of change of position, acceleration measures the rate of change of velocity. Whenever the velocity of a body changes — whether in magnitude, in direction, or in both — the body is said to be accelerating.

Average acceleration. If the velocity of an object changes from $v_1$ to $v_2$ over a time interval $\Delta t = t_2 - t_1$ , its average acceleration over that interval is defined as

\bar a = \frac{v_2 - v_1}{t_2 - t_1} = \frac{\Delta v}{\Delta t}

Instantaneous acceleration. Exactly as we passed from average velocity to instantaneous velocity by taking a limit, we define the instantaneous acceleration at time $t$ as

a(t) = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}

Since $v = dx/dt$ , acceleration can equally be written as the second derivative of position with respect to time,

a(t) = \frac{d}{dt}\left(\frac{dx}{dt}\right) = \frac{d^2x}{dt^2}

This is a second application of the same differentiation idea introduced in Section 2.4 — first to obtain velocity from position, then again to obtain acceleration from velocity.

Uniform and non-uniform acceleration. If the instantaneous acceleration of a body has the same constant value at every instant during a given interval, the motion is said to be uniformly accelerated over that interval; the kinematic equations of Section 2.8 apply only to this special (but very common and important) case. If the acceleration itself changes with time, the motion is said to have non-uniform acceleration, and the simple kinematic equations no longer apply directly — the more general calculus relations, $v = dx/dt$ and $a = dv/dt$ , remain valid in every case.

Sign convention: acceleration versus retardation. Once a positive direction has been fixed along the line of motion, acceleration — like velocity — carries an algebraic sign. It is important to realise that the sign of the acceleration by itself does not tell us whether the body is speeding up or slowing down; what matters is whether the acceleration is directed along the velocity or opposite to it:

If acceleration is in the same direction as the (instantaneous) velocity, the speed of the body increases with time.
If acceleration is opposite in direction to the velocity, the speed of the body decreases with time; this case is often called retardation or deceleration. A body braking to a halt (Example 5) has an acceleration opposite to its velocity, even though its velocity itself never changes sign before it stops. …

Misc 1Sign convention for acceleration and retardation

Worked out. A short explanatory schematic (not a numerical graph) showing a straight horizontal line representing the direction of motion, with the object's velocity vector drawn as an arrow pointing right (positive direction). Two cases are illustrated side by side: Case (i) acceleration arrow also pointing right, underneath labelled 'speeding up (positive acceleration)'; Case (ii) acceleration arrow pointing left, opposite to the velocity arrow, underneath labelled 'slowing down / retardation (negative acceleration, i.e. acceleration opposite to velocity)'. A short caption notes that the SIGN of acceleration alone does not tell us whether a body is speeding up or slowing down — what m …

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